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International Contests
Austrian-Polish
1988 Austrian-Polish Competition
2
\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0
\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0
Source: Austrian Polish 1988 APMC
April 30, 2020
inequalities
Sum
algebra
Problem Statement
If
a
1
≤
a
2
≤
.
.
≤
a
n
a_1 \le a_2 \le .. \le a_n
a
1
≤
a
2
≤
..
≤
a
n
are natural numbers (
n
≥
2
n \ge 2
n
≥
2
), show that the inequality
∑
i
=
1
n
a
i
x
i
2
+
2
∑
i
=
1
n
−
1
x
i
x
i
+
1
>
0
\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0
i
=
1
∑
n
a
i
x
i
2
+
2
i
=
1
∑
n
−
1
x
i
x
i
+
1
>
0
holds for all
n
n
n
-tuples
(
x
1
,
.
.
.
,
x
n
)
≠
(
0
,
.
.
.
,
0
)
(x_1,...,x_n) \ne (0,..., 0)
(
x
1
,
...
,
x
n
)
=
(
0
,
...
,
0
)
of real numbers if and only if
a
2
≥
2
a_2 \ge 2
a
2
≥
2
.
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